This article consist of a number of new soliton solutions for solving a complex nonlinear equation describing dynamics of nonlinear chains of atoms with long-range interactions via the ansatz function technique. In particular, we obtain lump, lump 1-kink, lump 2-kink, lump-periodic, periodic cross-rational (PCR), kink cross-rational (KCR) solutions, interaction of the multi-lump and periodic solutions as well as breather solutions. We also build rogue wave, generalized breather, homoclinic breather, Akhmediev breather, Ma breather, Kuznetsov-Ma breather, multiwave, M-shaped rational and some distinct interaction solutions. Mainly, for appropriate values of parameters, direct simulations have been performed to see the evolution of distinct solutions. More importantly, the solutions found in this work can have significant uses in Hamilton's equations and generalized momentum where solitons are used for long-range interactions.
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